Absolute Value Rules — Properties, Formula & Examples

Q: Can you distribute absolute value over addition or subtraction?

No. This is the single most common misconception. The absolute value of a sum is NOT the sum of the absolute values: $|a + b| \neq |a| + |b|$ in general. For example, $|3 + (-5)| = 2$, but $|3| + |-5| = 8$. You can only split absolute values over multiplication (product rule) and division (quotient rule).

Q: What is the difference between the triangle inequality and the alternate triangle inequality?

The triangle inequality gives an upper bound: $|a + b| \le |a| + |b|$. The alternate triangle inequality gives a lower bound: $|a - b| \ge \big||a| - |b|\big|$. Together they tell you that the absolute value of a sum or difference is trapped between these bounds. Both are used frequently in proofs and estimation problems.

Q: Why does $|-a| = |a|$?

Absolute value measures distance from zero on the number line. The numbers $a$ and $-a$ are the same distance from zero, just on opposite sides. Using the square root definition, $|-a| = \sqrt{(-a)^2} = \sqrt{a^2} = |a|$. This symmetry property means absolute value ignores the sign of the input.

Absolute Value Rules

Algebra rules for absolute values are listed below.

Piecewise Definition:	$\left\| a \right\| = \left\{ {\begin{array}{*{20}{c}}a&{{\rm{if}}\;a \ge 0}\\{ - a}&{{\rm{if}}\;a < 0}\end{array}} \right.$
Square root definition:	$\left\| a \right\| = \sqrt {{a^2}} $
Rules:	1. \|–a\| = \|a\| 2. \|a\| ≥ 0 3. Products: \|ab\| = \|a\|\|b\| 4. Quotients: \|a / b\| = \|a\| / \|b\| 5. Powers: \|aⁿ\| = \|a\|ⁿ 6. Triangle Inequality: \|a + b\| ≤ \|a\| + \|b\| 7. Alternate Triangle Inequality: \|a – b\| ≥ \|a\| – \|b\|
CAREFUL!!	Sums: Differences:	\|a + b\| is not the same as \|a\| + \|b\| \|a – b\| is not the same as \|a\| – \|b\|

Key Formula

\text{Piecewise: } |a| = \begin{cases} a & \text{if } a \ge 0 \\ -a & \text{if } a < 0 \end{cases}

\text{Product Rule: } |ab| = |a| \cdot |b|

\text{Quotient Rule: } \left|\frac{a}{b}\right| = \frac{|a|}{|b|}, \quad b \neq 0

\text{Triangle Inequality: } |a + b| \le |a| + |b|

Where:

$a$ = Any real number
$b$ = Any real number (nonzero for the quotient rule)
$|a|$ = The absolute value (distance from zero) of a

Worked Example

Problem: Simplify the expression

|(-3)(8)|

using the product rule, and verify the triangle inequality for

a = -3

and

b = 8

Step 1: Apply the product rule to split the absolute value over multiplication.

|(-3)(8)| = |-3| \cdot |8|

Step 2: Evaluate each absolute value separately. The absolute value of

-3

3

, and the absolute value of

8

8

= 3 \cdot 8 = 24

Step 3: You can verify by computing the product first:

(-3)(8) = -24

, and

|-24| = 24

. Both methods give the same result.

|{-24}| = 24 \checkmark

Step 4: Now check the triangle inequality with

a = -3

and

b = 8

. Compute the left side:

|a + b| = |-3 + 8| = |5| = 5

|{-3} + 8| = |5| = 5

Step 5: Compute the right side:

|a| + |b| = 3 + 8 = 11

. Since

5 \le 11

, the triangle inequality holds.

5 \le 3 + 8 = 11 \checkmark

Answer:

|(-3)(8)| = 24

, and the triangle inequality confirms

|{-3}+8| = 5 \le 11 = |{-3}|+|8|

Another Example

This example demonstrates the quotient rule (instead of the product rule) and verifies the alternate triangle inequality (reverse form), showing a different combination of rules.

Problem: Simplify

\left|\dfrac{-20}{4}\right|

using the quotient rule, and show that

|a - b| \ge |a| - |b|

for

a = -20

and

b = 4

Step 1: Apply the quotient rule to separate the absolute value over the fraction.

\left|\frac{-20}{4}\right| = \frac{|-20|}{|4|}

Step 2: Evaluate each absolute value:

|-20| = 20

and

|4| = 4

= \frac{20}{4} = 5

Step 3: Verify directly:

\frac{-20}{4} = -5

, and

|-5| = 5

. Same answer.

|-5| = 5 \checkmark

Step 4: Now test the alternate triangle inequality:

|a - b| \ge \big||a| - |b|\big|

. Compute the left side:

|-20 - 4| = |-24| = 24

|{-20} - 4| = |{-24}| = 24

Step 5: Compute the right side:

\big||{-20}| - |4|\big| = |20 - 4| = 16

. Since

24 \ge 16

, the inequality holds.

24 \ge |20 - 4| = 16 \checkmark

Answer:

\left|\dfrac{-20}{4}\right| = 5

, and the alternate triangle inequality gives

24 \ge 16

Frequently Asked Questions

Can you distribute absolute value over addition or subtraction?

No. This is the single most common misconception. The absolute value of a sum is NOT the sum of the absolute values:

|a + b| \neq |a| + |b|

in general. For example,

|3 + (-5)| = 2

, but

|3| + |-5| = 8

. You can only split absolute values over multiplication (product rule) and division (quotient rule).

What is the difference between the triangle inequality and the alternate triangle inequality?

The triangle inequality gives an upper bound:

|a + b| \le |a| + |b|

. The alternate triangle inequality gives a lower bound:

|a - b| \ge \big||a| - |b|\big|

. Together they tell you that the absolute value of a sum or difference is trapped between these bounds. Both are used frequently in proofs and estimation problems.

Why does

|-a| = |a|

Absolute value measures distance from zero on the number line. The numbers

a

and

-a

are the same distance from zero, just on opposite sides. Using the square root definition,

|-a| = \sqrt{(-a)^2} = \sqrt{a^2} = |a|

. This symmetry property means absolute value ignores the sign of the input.

Product/Quotient Rules (Equality) vs. Sum/Difference Rules (Inequality)

	Product/Quotient Rules (Equality)	Sum/Difference Rules (Inequality)
Statement	$\|ab\| = \|a\|\|b\|$ and $\|a/b\| = \|a\|/\|b\|$	$\|a+b\| \le \|a\|+\|b\|$ and $\|a-b\| \ge \|\|a\|-\|b\|\|$
Type of relation	Exact equality — always equal	Inequality — gives a bound, not an exact value
Can you split the absolute value?	Yes, you can freely split over × and ÷	No, you cannot split over + and −
Typical use	Simplifying products and fractions inside absolute values	Estimating or bounding sums and differences

Why It Matters

You encounter absolute value rules constantly when solving absolute value equations and inequalities in Algebra 1 and Algebra 2. In calculus and analysis, the triangle inequality is essential for proving limits, bounding error terms, and establishing convergence. Knowing which rules give equalities (products, quotients) versus inequalities (sums, differences) prevents errors in simplification that can derail an entire solution.

Common Mistakes

Mistake: Splitting absolute value over addition: writing

|a + b| = |a| + |b|

Correction: This is false in general. Absolute value distributes over multiplication and division, but NOT over addition or subtraction. For sums, you can only say

|a + b| \le |a| + |b|

(the triangle inequality). Try

a = 3, b = -5

|3 + (-5)| = 2

, but

|3| + |-5| = 8

Mistake: Assuming

|a| = a

without checking the sign of

a

Correction: By the piecewise definition,

|a| = a

only when

a \ge 0

. If

a < 0

, then

|a| = -a

(which is positive). For instance, if

a = -7

, then

|a| = -(-7) = 7

, not

-7

. Always consider both cases when working with variables.

Related Terms

Absolute Value — Core definition that these rules apply to
Triangle Inequality with Absolute Value — Key inequality rule for sums
Piecewise Function — Used in the piecewise definition of absolute value
Square Root — Used in the square root definition $|a| = \sqrt{a^2}$
Algebra — Broader subject where these rules are applied
Absolute Value of a Complex Number — Extends absolute value rules to complex numbers
Triangle Inequality — General geometric form of the inequality
Sum — Operation where absolute value does NOT distribute

Piecewise Definition:	\(\left\| a \right\| = \left\{ {\begin{array}{*{20}{c}}a&{{\rm{if}}\;a \ge 0}\\{ - a}&{{\rm{if}}\;a < 0}\end{array}} \right.\)
Square root definition:	\(\left\| a \right\| = \sqrt {{a^2}} \)
Rules:	1. \|–a\| = \|a\| 2. \|a\| ≥ 0 3. Products: \|ab\| = \|a\|\|b\| 4. Quotients: \|a / b\| = \|a\| / \|b\| 5. Powers: \|aⁿ\| = \|a\|ⁿ 6. Triangle Inequality: \|a + b\| ≤ \|a\| + \|b\| 7. Alternate Triangle Inequality: \|a – b\| ≥ \|a\| – \|b\|
CAREFUL!!	Sums: Differences:	\|a + b\| is not the same as \|a\| + \|b\| \|a – b\| is not the same as \|a\| – \|b\|